58 research outputs found
Elements of a Theory of Simulation
Unlike computation or the numerical analysis of differential equations,
simulation does not have a well established conceptual and mathematical
foundation. Simulation is an arguable unique union of modeling and computation.
However, simulation also qualifies as a separate species of system
representation with its own motivations, characteristics, and implications.
This work outlines how simulation can be rooted in mathematics and shows which
properties some of the elements of such a mathematical framework has. The
properties of simulation are described and analyzed in terms of properties of
dynamical systems. It is shown how and why a simulation produces emergent
behavior and why the analysis of the dynamics of the system being simulated
always is an analysis of emergent phenomena. A notion of a universal simulator
and the definition of simulatability is proposed. This allows a description of
conditions under which simulations can distribute update functions over system
components, thereby determining simulatability. The connection between the
notion of simulatability and the notion of computability is defined and the
concepts are distinguished. The basis of practical detection methods for
determining effectively non-simulatable systems in practice is presented. The
conceptual framework is illustrated through examples from molecular
self-assembly end engineering.Comment: Also available via http://studguppy.tsasa.lanl.gov/research_team/
Keywords: simulatability, computability, dynamics, emergence, system
representation, universal simulato
Order Independence in Asynchronous Cellular Automata
A sequential dynamical system, or SDS, consists of an undirected graph Y, a
vertex-indexed list of local functions F_Y, and a permutation pi of the vertex
set (or more generally, a word w over the vertex set) that describes the order
in which these local functions are to be applied. In this article we
investigate the special case where Y is a circular graph with n vertices and
all of the local functions are identical. The 256 possible local functions are
known as Wolfram rules and the resulting sequential dynamical systems are
called finite asynchronous elementary cellular automata, or ACAs, since they
resemble classical elementary cellular automata, but with the important
distinction that the vertex functions are applied sequentially rather than in
parallel. An ACA is said to be pi-independent if the set of periodic states
does not depend on the choice of pi, and our main result is that for all n>3
exactly 104 of the 256 Wolfram rules give rise to a pi-independent ACA. In 2005
Hansson, Mortveit and Reidys classified the 11 symmetric Wolfram rules with
this property. In addition to reproving and extending this earlier result, our
proofs of pi-independence also provide significant insight into the dynamics of
these systems.Comment: 18 pages. New version distinguishes between functions that are
pi-independent but not w-independen
Cycle Equivalence of Graph Dynamical Systems
Graph dynamical systems (GDSs) can be used to describe a wide range of
distributed, nonlinear phenomena. In this paper we characterize cycle
equivalence of a class of finite GDSs called sequential dynamical systems SDSs.
In general, two finite GDSs are cycle equivalent if their periodic orbits are
isomorphic as directed graphs. Sequential dynamical systems may be thought of
as generalized cellular automata, and use an update order to construct the
dynamical system map.
The main result of this paper is a characterization of cycle equivalence in
terms of shifts and reflections of the SDS update order. We construct two
graphs C(Y) and D(Y) whose components describe update orders that give rise to
cycle equivalent SDSs. The number of components in C(Y) and D(Y) is an upper
bound for the number of cycle equivalence classes one can obtain, and we
enumerate these quantities through a recursion relation for several graph
classes. The components of these graphs encode dynamical neutrality, the
component sizes represent periodic orbit structural stability, and the number
of components can be viewed as a system complexity measure
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